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Mirrors > Home > MPE Home > Th. List > csbima12 | Structured version Visualization version Unicode version |
Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.) |
Ref | Expression |
---|---|
csbima12 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3352 |
. . . 4
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2 | csbeq1 3352 |
. . . . 5
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3 | csbeq1 3352 |
. . . . 5
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4 | 2, 3 | imaeq12d 5175 |
. . . 4
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5 | 1, 4 | eqeq12d 2486 |
. . 3
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6 | vex 3034 |
. . . 4
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7 | nfcsb1v 3365 |
. . . . 5
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8 | nfcsb1v 3365 |
. . . . 5
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9 | 7, 8 | nfima 5182 |
. . . 4
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10 | csbeq1a 3358 |
. . . . 5
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11 | csbeq1a 3358 |
. . . . 5
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12 | 10, 11 | imaeq12d 5175 |
. . . 4
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13 | 6, 9, 12 | csbief 3374 |
. . 3
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14 | 5, 13 | vtoclg 3093 |
. 2
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15 | csbprc 3774 |
. . 3
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16 | csbprc 3774 |
. . . . 5
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17 | 16 | imaeq2d 5174 |
. . . 4
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18 | ima0 5189 |
. . . 4
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19 | 17, 18 | syl6req 2522 |
. . 3
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20 | 15, 19 | eqtrd 2505 |
. 2
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21 | 14, 20 | pm2.61i 169 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-fal 1458 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-br 4396 df-opab 4455 df-xp 4845 df-cnv 4847 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 |
This theorem is referenced by: csbrn 5304 disjpreima 28271 csbpredg 31797 brtrclfv2 36390 sbcheg 36445 |
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